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MA2223: PROBLEM SHEET 1 1. Show that the diameter of an open ball B(a, r) in Euclidean space Rn is exactly 2r. 2. Let C[0, 1] be the set of continuous real-valued functions on the interval [0, 1] together with the supremum metric d(f, g) = sup |f (x) − g(x)| x∈[0,1] Let f ∈ C[0, 1] be the function f (x) = x. Sketch the graph of f and the region which contains the graphs of all elements of the open ball B(f, 1). 3. Let (X, d) be a metric space and let A be a subset of X. The set of all interior points of A is called the interior of A and is denoted A◦ . (a) Prove that A◦ is an open set in (X, d). (b) Prove that for any subset B we have (A ∩ B)◦ = A◦ ∩ B ◦ . 4. Let X be a non-empty set and let d : X × X → R satisfy the following conditions (a) d(x, y) = 0 if and only if x = y, (b) d(x, z) ≤ d(y, x) + d(z, y) for all x, y, z ∈ X. Show that d is a metric on X. 1